Theorem u4lemnob 655

Description: Lemma for non-tollens implication study.

Assertion

Ref	Expression
u4lemnob	((a ->4 b)_|_ v b) = ((a ^ b_|_) v b)

Proof of Theorem u4lemnob

Step	Hyp	Ref	Expression
1		u4lemanb 600	. . . 4 ((a ->4 b) ^ b_|_) = ((a_|_ v b) ^ b_|_)
2		oran2 84	. . . . 5 (a_|_ v b) = (a ^ b_|_)_|_
3	2	ran 71	. . . 4 ((a_|_ v b) ^ b_|_) = ((a ^ b_|_)_|_ ^ b_|_)
4	1, 3	ax-r2 35	. . 3 ((a ->4 b) ^ b_|_) = ((a ^ b_|_)_|_ ^ b_|_)
5		anor1 80	. . 3 ((a ->4 b) ^ b_|_) = ((a ->4 b)_|_ v b)_|_
6		anor3 82	. . 3 ((a ^ b_|_)_|_ ^ b_|_) = ((a ^ b_|_) v b)_|_
7	4, 5, 6	3tr2 61	. 2 ((a ->4 b)_|_ v b)_|_ = ((a ^ b_|_) v b)_|_
8	7	con1 63	1 ((a ->4 b)_|_ v b) = ((a ^ b_|_) v b)

Syntax hints:   = wb 1  _|_wn 4   v wo 6   ^ wa 7   ->4 wi4 16

This theorem was proved from axioms:  ax-a1 29  ax-a2 30  ax-a3 31  ax-a4 32  ax-a5 33  ax-r1 34  ax-r2 35  ax-r4 36  ax-r5 37  ax-r3 421

This theorem depends on definitions:  df-b 38  df-a 39  df-t 40  df-f 41  df-i4 46  df-le1 122  df-le2 123  df-c1 124  df-c2 125

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